All you need to do is build a rocket and have large quantities of food and water as well as life support systems. Then when the war starts, launch yourself towards mars with some other people and hope to start a colony. :D If that doesn't work make a massive underground nuclear fallout shelter and live there forever.

. Resistance to radiation is only one part of the equation.. Quantity - there are a lot of cockroaches. "There are about 4,000 species of cockroach..."1. Distribution - cockroaches are everywhere. "Cockroaches live in a wide range of environments around the world."1. Environment - cockroaches tend to hang out in protected places.. Hardiness - "Some species are capable of remaining active for a month without food ... Some can go without air for 45 minutes ... "1. . While not the most radiation-resistant insect, the cockroach does stand a good chance of inheriting the Earth. Looks like the common fruit fly is in the running. Most mammals are out.